3. Quantitative Foundations

Question 1

What is the general term denoting compound interest when the interest is not continuously compounded?

Answer 1

Discrete compounding

Question 2

What is the primary challenge that causes difficulty in calculating the return performance of a forward contract or other position that requires no net investment? How is that challenge addressed?

Answer 2

If the forward contract has a starting value of zero, it would cause division by zero. One solution to the problem of computing return for derivatives is to base the return on notional principal. Another is to include collateral.

Question 3

Consider a position in a single forward contract. What distinguishes a fully collateralized position in this forward contract from a partially collateralised position?

Answer 3

A fully collateralized position is paired with a quantity of capital equal in value to the notional principal of the contract whereas a partially collateralized position is paired with collateral lower in value than the notional value.

Question 4

An IRR is estimated for a fund based on an initial investment when the fund was created, several annual distributions and an estimate of the fund’s value prior to its termination. What type of IRR is this?

Answer 4

Since Inception IRR

Question 5

An investment has two solutions for its IRR. What can be said about the investment and the usefulness of the two solutions?

Answer 5

There are two sign changes in the cash flow stream of the investment. None of the IRRs should be used.

Question 6

Two investments are being compared to ascertain which investment would add the most value to a portfolio. Both investments have simplified cash flow patterns of an initial cost followed by positive cash flows. Why might the IRRs of the investment provide an unreliable indication of which investment adds more value?

Answer 6

The major challenge with comparing IRRs across investments is when investments have scale differences. Scale differences are when investments have unequal sizes and/or timing of their cash flows.

Question 7

An analyst computes the IRR of one alternative to be 20% and another to be 30%. When the analyst combines the cash flows of the two alternatives into a single investment, must the IRR of the combination be greater than 20% and less than 30%?

Answer 7

No. The answer is not immediately apparent because the IRR of a portfolio of two investments is not generally equal to a value-weighted average of the IRRs of the constituent investments. If the cash flows from two investments are combined to form a portfolio, the IRR of the portfolio can vary substantially from an average of the IRRs of the two investments.

Question 8

Is an IRR a dollar-weighted return or a time-weighted return? Why?

Answer 8

The IRR is the primary method of computing a dollar-weighted return.

The time-weighted rate of return (TWR) is a measure of the compound rate of growth in a portfolio. The TWR measure is often used to compare the returns of investment managers because it eliminates the distorting effects on growth rates created by inflows and outflows of money.

The money-weighted rate of return is calculated by finding the rate of return that will set the present values of all cash flows equal to the value of the initial investment.

Question 9

In which scenario will a clawback clause lead to payments?

Answer 9

A clawback clause, clawback provision or clawback option is designed to return incentive fees to LPs when early profits are followed by subsequent losses. A clawback provision requires the GP to return cash to the LPs to the extent that the GP has received more than the agreed profit split. A GP clawback option ensures that if a fund experiences strong performance early in its life and weaker performance at the end, the LPs get back any incentive fees until their capital contributions, expenses, and any preferred return promised in the partnership agreement have been paid.

Question 10

What is the difference between a hard hurdle rate and a soft hurdle rate?

Answer 10

A hard hurdle rate limits incentive fees to profits in excess of the hurdle rate. A soft hurdle rate allows fund managers to earn an incentive fee on all profits, given that the hurdle rate has been achieved.

Question 11

What is the difference between simple interest, continuous compounding?

Answer 11

Simple interest: is an interest rate computation approach that does not incorporate compounding. Continuous compounding assumes that earnings can be instantaneously reinvested to generate additional earnings.

Question 12

The rate of return that discounts a value of $110 to be received in the future to a present value of $100 expressed as a total (non-annualized) rate is 0.10.

What is the lognormal return ?

Answer 12

Logarithmic return = ln( 1 +R ) –> ln( 1.1 ) = 0.0953 or 9.53 %

Question 13

What are the three other IRRs based on the time after since-inception IRRs?

Answer 13

1. Lifetime IRRs: a lifetime IRR contains all of the cash flows, realised or anticipated, occurring over the investment’s entire life, from period o to period T. 2. Interim IRRs: the interim IRR is a computation of IRR based on realised cash flows from an investment and its current estimated residual value. 3. Point-to-point IRR: is a calculation of performance over part of an investment’s life. All cash flows are based on realised or appraised values rather than expected cash flow over the investment’s projected life. Although any IRR is calculated from one point in time to another, a point-to-point IRR would typically not be used to refer to a lifetime iRR.

Question 14

Investment A is expected to cost $100 and to be followed by cash inflows of $10 after one year and then $120 after the second year, when the project terminates. The IRR is based on the anticipated cash flows and is an anticipated lifetime IRR.

Answer 14

2 N 100 +/- PV 10 PMT 120 - 10 = 110 FV CPT I/Y = 14.65%

Question 15

Fund B expended $200 million to purchase investments and distributed $30 million after one year. At the end of the second year, it is being appraised at $180million.

Answer 15

2 N 200,000,000 +/- PV 30,000,000 PMT 180,000,000 - 30,000,000 150,000,000 FV CPT I/Y = 2.66%

Question 16

Investment C had been in existence three years when it was purchased by BK fund for $500. In the three years following the purchase, the investment distributed cash flows to the investor of $110, $120, and $130. Now in the forth year, the investment has been appraised as being worth $400. The IRR is based on realized cash flows and an appraised value.

Answer 16

CF 2nd CE/C 500 +/- Enter \/ 110 Enter \/ \/ 120 Enter \/ \/ 130 Enter \/ \/ 400 Enter IRR CPT = 15.05%

Question 17

What is the difference between time-weighted and dollar-weighted returns?

Answer 17

Time-weighted returns are averaged returns that assume that no cash was contributed or withdrawn during the averaging period, meaning after the initial investment. Dollar weighted are averaged returns that are adjusted for the therefore reflect when cash has been contributed or withdrawn during the averaging period.

Question 18

What is the difference between a catch-up provision and hurdle rate?

Answer 18

A hurdle rate specifies a return level that LPs must receive before GPs begin to receive incentive fees. Whereas, the catch-up rate provision permits the fund manager to receive a large share of profits once the hurdle rate of return has been achieved and passed.

Question 19

Fund A at the end of its term has risen to a total net asset value (NAV) of $300 million from its initial size of $200 million. Assuming no hurdle rate and an 80%/20% carried-interest split, the general partner is entitled to receive carried interest equal to how much?

Answer 19

20 million for the GP

Question 20

Fund B terminates and ultimately returns $132 million to its limited partners, and the total initial size of the fund was $100 million. Assuming a carried-interest rate of 20% the general partner is entitled to receive carried interest equal to how much?

Answer 20

$132 is the limited partners return. Which if the carried-interest rate is 20%, means the limited partners received 80%. With a starting of $100 million. The difference being 32, divided by 0.8% = 40. Minus the 32 = 8. 8 million being the GP return.

Question 21

Consider a fund that makes two investments, A and B, of $10 million each. Investment A is successful and generates a $10 million profit, whereas Investment B is a complete write-off (a total loss). Assume that the fund managers are allowed to take 20% of profits as carried interest. How much carried interest will they receive if profits are calculated on a fund-asa- whole (aggregated) basis, and how much will they receive if profits are calculated on a deal-by-deal (individual transaction) basis?

Answer 21

On the fund-as-a-whole basis, the fund broke even, so no incentive fees will be distributed. On the deal-by-deal basis, Investment A earned $10 million, so $2 million in carried interest will be distributed to the managers.

Question 22

Consider a fund that calculates incentive fees on a fund-as-a-whole basis and makes two investments, A and B, of $10 million each. Investment A is successful and generates a $10 million profit after three years. Investment B is not revalued until it is completely written off after five years. Assume that the fund managers are allowed to take 20% of profits as carried interest calculated on an aggregated basis. How much carried interest will they receive if there is no clawback provision, and how much will they receive if there is a clawback provision?

Answer 22

Without a clawback provision, the fund earned $10 million after three years and distributed a $2 million carried interest to the managers. When the second investment failed, the incentive fee is not returned. In the case of a clawback provision, the fund distributed a $2 million incentive fee to the managers after three years, but when the second investment failed, the incentive fee is returned to the limited partners, since there is no combined profit.

Question 23

Consider a $10 million fund with a 20% incentive fee that lasts a single year and earns a $2 million profit. With a hard hurdle rate of 10%.

Answer 23

Ignoring a hurdle rate, the fund manager would receive $400,000, which is 20% of $2 million. But with a hard hurdle rate of 10%, the fund manager receives the 20% incentive fees only on profits in excess of the 10% return, meaning $200,000. The first $1 million of profit goes directly to the limited partners. The fund manager collects an incentive fee only on profits in excess of the $1 million, which is the profit necessary to bring the limited partners’ return up to the hurdle rate. Thus, the manager receives an incentive fee of $200,000.

Question 24

Fund A with an initial investment of $20 million liquidates with $24 million cash after one year. The hurdle rate is 15%, and the incentive fee is 20%. What is the distribution to the fund manager if the fund uses a hard hurdle? What is the distribution to the fund manager if the fund has a soft hurdle and a 50% catch-up rate?

Answer 24

The first $20 million is returned to the limited partners in both cases. With a hard hurdle, the limited partners receive the first $3 million of profit, which is 15% of the $20 million investment. The fund manager receives 20% of the remaining profit of $1 million, which is $200,000. The limited partners receive 80% of the remaining $1 million, which is $800,000, for a total profit of $3.8 million. With a soft hurdle, the limited partners receive the first $3 million of profit, which is 15% of the $20 million investment. To fulfill the catch-up provision, the fund manager receives 50% of the remaining profit up to the point of being paid 20% of all profit. In this case, 50% of all of the remaining profit, or $1 million, is $500,000. Since $500,000 is less than 20% of the entire $4 million profit, the fund manager is unable to fully catch up. Had the total profits exceeded $5 million, the catch-up of the fund manager would have been completed. With $5 million of profit, the GP would receive 50% of the profits above $3 million, or $1 million (50% of the $2 million profit in excess of the profit necessary to meet the hurdle rate for the LPs). The $1 million of catch-up equals 20% of $5 million. Profits in excess of $5 million would then be split 20% to the fund manager and 80% to the limited partners

Question 25

What is the hard hurdle sequence?

Answer 25

1. Capital is returned to the limited partners until their investment has been repaid. 2. Profits are distributed only to the limited partners until the hurdle rate is reached. 3. Additional profits are split such that the fund manager receives an incentive fee only on the profits in excess of the hurdle rate.

Question 26

The soft hurdle sequence is?

Answer 26

1. Capital is returned to the limited partners until their investment has been repaid. 2. Profits are distributed only to the limited partners until the hurdle rate reached. 3. Additional profits are split, with a high proportion going to the fund manager until the fund manager receives an incentive fee on all of the profits.